I've been struggling against the below deduction of the approximation.
$0<a\in \mathbb{R}$
$d:=\sqrt{x^2+y^2} \hspace{0.3cm}$ where $ \hspace{0.1cm} 0\leq x,y\in \mathbb{R}$
$a\ll d$
$$\frac{1}{\sqrt{1-\frac{2ax}{d^2}}}+\frac{1}{\sqrt{1+\frac{2ax}{d^2}}}\simeq \left(\left( 1+\frac{ax}{d^2}\right)-\left(1-\frac{ax}{d^2}\right)\right)$$
Can anyone tell me some hint(s) so that I can deduce it myself?
Hint:
Can you show $\frac{2ax}{d^2} \ll 1$?
If so, then since $(1+\epsilon)^{-1/2} = 1 -\frac{\epsilon}{2} + O(\epsilon^3)$, you can compute
$$ \frac{1}{\sqrt{1+\frac{2ax}{d^2}}} = \left ( 1 + \frac{2ax}{d^2} \right )^{-1/2} = 1 - \frac{ax}{d^2} + O \left ( \left ( \frac{2ax}{d^2} \right )^3 \right ) $$
And similarly for $\left ( 1 - \frac{2ax}{d^2} \right )^{-1/2}$.
Do you see how to finish the problem from here?
I hope this helps ^_^
